Theorems of Katznelson-Tzafriri type for semigroups of operators

Dissipative operators on Banach spaces

“…In the proof of the next lemma, we follow basically renorming technique, which had been used previously by Esterle [10], Esterle et al [11], Phong [21], Allan and Ransford [1] and others. Then, p is a seminorm on X.…”

“…It follows from Theorem 4.4 that if T ∈ B(X) is a dissipative operator with zero unitary spectrum, then lim t→∞ e tT T = 0 (see, also [15,21]). In [25] Swiech proved that the set {T n , n = 1, 2, .…”

The Banach algebra generated by a -semigroup

“…Recall [5, p. 163] that the function f ∈ L 1 (R + ) is said to be of spectral synthesis with respect to a closed subset K of R if it can be approximated (in L 1 -norm) by functions g n ∈ L 1 (R) such thatĝ n = 0 on a neighborhood of K. Semigroup version of the Katznelson-Tzafriri Theorem (see, [3,6] and [5,Theorem 5…”

On the Decay of Bounded Semigroup on the Domain of its Generator

“…At the same time, Batty [2,3] and Vu Phong [10] proved the following result (formulation is taken from [5]). …”